The numerical quantization method is a grid method that relies on the approximation of the solution to a nonlinear problem by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first-order scheme based on piecewise linear approximations. Main ingredients are correction terms in the transition probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and a priori estimates of the errors are proposed. | 
